Before presenting the evolutionary outcomes of spatially structured populations we note that in a well-mixed system cooperators cannot survive if \(T>1\)  in the prisoner's dilemma game or if \(r<G\)  in the public goods game where \(G\)  denotes the actual size of the group. On square lattice the critical temptation value is \(T=1.0655\)  where cooperators die out in the homogeneous prisoner's dilemma game and the minimal multiplication factor is \(r=3.74\)  which is necessary for cooperators to survive in the public goods game [1]}, [2]}. Importantly, if we apply the earlier mentioned \(\alpha =0.266\)  weighting factor for the public goods payoff values then this critical \(r\)  value shifts to \(r=4.33\) .
<FIGURE>